2014/04/22 by Hoài-Minh Nguyên, Linh V. Nguyen, Nguyen, Hoai-Minh +1
Engineering · Physics and Astronomy · #Analysis of PDEs (math.AP) #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1404.5580
openalex publication_date 2014/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is devoted to the study of the generalized impedance boundary conditions (GIBCs) for a strongly absorbing obstacle in the \bf time regime in two and three dimensions. The GIBCs in the time domain are heuristically derived from the corresponding conditions in the time harmonic regime. The latters are frequency dependent except the one of order 0; hence the formers are non-local in time in general. The error estimates in the time regime can be derived from the ones in the time harmonic regime when the frequency dependence is well-controlled. This idea is originally due to Nguyen and Vogelius in \citeNguyenVogelius2 for the cloaking context. In this paper, we present the analysis to the GIBCs of orders 0 and 1. To implement the ideas in \citeNguyenVogelius2, we revise and extend the work of Haddar, Joly, and Nguyen in \citeHJNg1, where the GIBCs were investigated for a fixed frequency in three dimensions. Even though we heavily follow the strategy in \citeNguyenVogelius2, our analysis on the stability contains new ingredients and ideas. First, instead of considering the difference between solutions of the exact model and the approximate model, we consider the difference between their derivatives in time. This simple idea helps us to avoid the machinery used in \citeNguyenVogelius2 concerning the integrability with respect to frequency in the low frequency regime. Second, in the high frequency regime, the Morawetz multiplier technique used in \citeNguyenVogelius2 does not fit directly in our setting. Our proof makes use of a result by Hörmander in \citeHor. Another important part of the analysis in this paper is the well-posedness in the time domain for the approximate problems imposed with GIBCs on the boundary of the obstacle, which are non-local in time.